Course in Probability Theory

7 .2 LINDEBERG-FELLER THEOREM 1 223*7. Find an example where Lindeberg's condition is satisfied butLiapounov's is not for any 8 > 0 .In Exercises 8 to 10 below {X j , j > 1 } is a sequence of independent r .v .'s .8. For each j let X j have the uniform distribution in [-j, j] . Show thatLindeberg's condition is satisfied and state the resulting central limit theorem .9. Let X,. be defined as follows for some a > 1 :1fja , with probability 6 j2(a_1) each ;X~ - 10, with probability 1 - 3 j2(a-1)Prove that Lindeberg's condition is satisfied if and only if a < 3/2 .* 10 . It is important to realize that the failure of Lindeberg's conditionmeans only the failure of either (i) or (ii) in Theorem 7 .2 .1 with the specifiedconstants s, A central limit theorem may well hold with a different sequenceof constants . Letwith probability each ;12j 2with probability 12 each ;with probability 1 - 1 - 16 6j 2Prove that Lindeberg's condition is not satisfied . Nonetheless if we take b, _11 3/18, then S„ /b„ converges in d ist . to (D . The point is that abnormally largevalues may not count! [HINT : Truncate out the abnormal value .]11 . Prove that fo x 2 d F (x) < oo implies the condition (11), but notvice versa .* 12 . The following combinatorial problem is similar to that of the numberof inversions . Let Q and '~P be as in the example in the text . It is standardknowledge that each permutation1a,2a2can be uniquely decomposed into the product of cycles, as follows . Considerthe permutation as a mapping n from the set (1, . . ., n) onto itself suchthat 7r(j) = a j . Beginning with 1 and applying the mapping successively,1 --* 7r (1) - 2r 2 (1) -* . . ., until the first k such that rr k (1) = 1 . Thus(1, 7r(l), 7r2(1), . . . , 7rk-1(1))